---

Subject and independence

The reviewer is an independent examiner working in a fresh context from the commissioned assignment alone and took no part in writing the page, the neighboring reconstruction pages, or the library card. The charge is refutation.

Subject: path wiki/research/erdos_501/glazer_theorem_5_1_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read whole as of that time.

Artifact: the PDF held by the library card Glazer (2026) (draft rev10, eight pages, printed page numbers equal to physical page numbers). The text layer of all eight pages was extracted; Theorem 5.1 and its proof at physical pp. 6--7 were read clause by clause, every display (5.1)--(5.11) checked against the page images. Page images were rendered for all eight pages at 130 dpi and pages 3, 4, 5, 6 and 7 were read as images: p. 3 for the coding conventions and Definition 3.1 (displays (3.1)--(3.4)), p. 4 for the opening of Section 4 and Lemma 4.1, p. 5 for Proposition 4.4 and display (4.2), p. 6 for Lemma 4.5, display (4.3) and the start of Theorem 5.1, and p. 7 for the rest of the proof.

Allowed material actually read: the page; the cited reconstruction pages glazer_theorem_3_2_reconstruction, glazer_proposition_4_4_reconstruction, glazer_lemma_4_5_reconstruction and glazer_lemma_4_1_reconstruction as of the same time; the Theorem 1.1 page linked from the page's Boundary paragraph; the library card's provenance paragraph; the problem page wiki/problems/set_theory/E0501/_index.md from its heading to the line before "Current assessment" (the page has no "Statement" heading; its Statement paragraph sits in that span); docs/verification.md "Whole-claim report" and both "Audit checklist" sections; docs/evidence.md "Source fidelity"; and docs/math_authoring.md.

Exposures: three, all through whole-file printing rather than intent. First, git show printed the entire library card, so the reviewer saw its "Read status", "Companion formalization" and "Relation to E501" paragraphs, the last carrying acceptance text. Second, the E0501 span read for the Statement also contains that page's "Status" paragraph. Third, the four cited reconstruction pages and the Theorem 1.1 page came through whole, so their Proof and Boundary sections were seen as well as their Statement sections; the Lemma 4.5 and Proposition 4.4 proofs were then used only to confirm the interfaces (the pullback convention, the "uncountable in the ground model" hypothesis). None of the exposed text was used in any mathematical check below, and no Current assessment, no Known results, no other review, nothing under any evidence/ folder other than this report's own path, and nothing outside the repository was read. No web search was made.

Restatement

The following is a theorem of ZFC + CH about the forcing relation. Let $\kappa=\omega_2$, $\Theta=\kappa\times\omega$, and let $\mathbb B$ be the measure algebra of the completed product of fair-coin measures on $2^\Theta$ (the source's $\mathbb B_{\omega_2}$). Then the top condition of $\mathbb B$ forces the following. For every family $\mathcal A=(A_y)_{y\in\mathbb R}$ of subsets of $\mathbb R$, that is, every function from the reals of the extension to sets of those reals: if the Lebesgue outer measure $\lambda^*(A_y)$ is below one for every $y\in\mathbb R$ (boundedness is not assumed), then a profile certificate for $\mathcal A$ exists, namely

• a standard Borel probability space $(\Omega,\nu)$ and a set $Z\subseteq\Omega$, not required to be measurable, with $\nu^*(Z)=1$ where $\nu^*$ is the infimum of $\nu$ over Borel supersets;
• for every $m\in\mathbb Z$ a Borel map $x_m\colon\Omega\to[m,m+1)$ with $\nu(x_m^{-1}(B))=\lambda(B\cap[m,m+1))$ for every Borel $B\subseteq\mathbb R$;
• for every $m\in\mathbb Z$ a Borel map $c_m\colon\Omega\to\mathcal O$ into a standard Borel coding space of open subsets of $\mathbb R$, for which $x\in U(c)$ and $c\mapsto\lambda(U(c))$ are Borel, with $\lambda(U(c_m(z)))<1$ for every $z\in\Omega$;
• $A_{x_m(z)}\subseteq U(c_m(z))$ for every $z\in Z$ and every $m\in\mathbb Z$.

Conventions carried by the page: the outer measure $\lambda^*(A)$ is the infimum of $\lambda(U)$ over open $U\supseteq A$; the pullback of $v\in2^{P'}$ along a bijection $\pi\colon P\to P'$ is written $v\circ\pi$ where the source writes $\pi^{-1}(v)$; $2^{\{\alpha\}\times\omega}$ and $2^D$ are identified with $2^\omega$ through the enumerations $n\mapsto(\alpha,n)$ and $\langle d_n\rangle$; and $\rho$ is any Borel map $2^\omega\to[0,1)$ carrying fair-coin measure to Lebesgue measure with null point fibers. The page claims only an author-recorded reconstruction: no tier, no status change.

Checklist

• Quantifiers and scope. Pass. The page keeps the source's quantifiers in place: $\forall\mathcal A$ over families, $\forall y\in\mathbb R$ in the hypothesis, $\forall\alpha<\kappa$ and $\forall m\in\mathbb Z$ for the envelope names, (P2) and (P3) for every $z\in\Omega$, (P4) for every $z\in Z$. Nothing "almost all" is upgraded to "all": the one almost-everywhere phenomenon (codes of measure at least one off $Z$) is removed by the truncation exactly as in the source. The boundary case $p=1$ makes the mixing empty and is harmless.
• Circularity. Pass. The inputs are Proposition 4.4 and Lemma 4.5, neither of which consumes Theorem 5.1; Theorem 3.2 is not used on the page.
• Model and convention changes. Pass. The pullback convention, the two identifications with $2^\omega$, and outer regularity taken as the definition of $\lambda^*$ are declared and agree with the source's $\pi_\alpha^{-1}(G\restriction P_\alpha)$ and "outer regularity". Lemma 4.5 is applied with $\Gamma=\Theta\setminus\bigcup_{\alpha\in J}P_\alpha$, so the forcing it speaks about is $\mathbb B(\Theta)$ itself, the algebra of the theorem.
• Finite and statistical overreach. Inapplicable: no finite case, sample or heuristic appears.
• Uniformity. Pass. The only uniformity in the argument, one Borel map $F$ reading every $\dot w_\alpha$ for $\alpha\in J$, is delivered by Proposition 4.4's interface at exactly that strength; the constant $1$ in $\lambda(U(c))<1$ is the source's; no limit or sum is exchanged.
• Extremal conclusions. Pass. $\nu^*(Z)=1$ is imported at exactly the strength of Lemma 4.5, with $\nu^*$ as defined on the Theorem 3.2 page.
• Consequences and composition. Pass. Every "hence" was re-derived: (5.4) from the maximum principle; $\Vdash\dot c_{\alpha,m}\in\mathcal O$ from the mixing; (P1) from Lemma 4.5; (P3) from the truncation; (P2) from the two pushforwards; $\dot c_{\alpha,m}^G=c^0_m(z_\alpha)$ and $x_m(z_\alpha)=\dot x_{\alpha,m}^G$ from (4.2) and the choice of $\pi_\alpha$; (5.11) from (5.4) with $p\in G$; and the closing passage to the top condition. Details are under Weakest steps.
• Computation. Inapplicable: the page carries no computation.
• Reproduction. Inapplicable: the page states no rerun command or coverage claim.
• Source and verdict fidelity. Pass. The Statement matches (5.1); the labels (5.1)--(5.11) each name the display the page attributes to them; the locator "physical pp. 6--7" is right; the Standing paragraph claims author-recorded standing and nothing more.

Weakest steps

W1. Transfer of the forced envelope (5.4) to (P4) at $z_\alpha$. In $M[G]$ the reading (4.2), forced by the top condition, gives $\langle\dot c_{\alpha,m}^G:m\in\mathbb Z\rangle=\dot w_\alpha^G=F(g,z_\alpha)$, so $\dot c_{\alpha,m}^G=F_m(g,z_\alpha)=c^0_m(z_\alpha)$. For $n<\omega$, $z_\alpha(d_n)=u_G(\pi_\alpha(d_n))=u_G(\alpha,n)=\dot r_\alpha^G(n)$, because $\pi_\alpha(d_n)=(\alpha,n)\in D_\alpha\subseteq P_\alpha$; so under the two fixed identifications $z_\alpha\restriction D$ and $\dot r_\alpha^G$ are the same element of $2^\omega$, and $x_m(z_\alpha)=m+\rho(\dot r_\alpha^G)=\dot x_{\alpha,m}^G$, with $\rho$ reinterpreted from its code by (R4). Since $p\in G$ and the mixed name agrees with the original below $p$, (5.4) holds in $M[G]$: $A_{x_m(z_\alpha)}\subseteq U(c^0_m(z_\alpha))$ and $\lambda(U(c^0_m(z_\alpha)))<1$. The inequality places $z_\alpha$ in the untruncated case of (5.8), so $c_m(z_\alpha)=c^0_m(z_\alpha)$ and (5.11) follows. This composes with the rest as (P4); it is the only place where $p\in G$ is needed.

W2. The truncation and (P3). $c^0_m=F_m(g,\cdot)$ is a section of the Borel map $F_m$ at the point $g$ and so is Borel in $M[G]$. The set $B_m=\{z\in\Omega:\lambda(U(c^0_m(z)))<1\}$ is the preimage of $[0,1)$ under the Borel map $\lambda\circ U\circ c^0_m$, hence Borel. The map $c_m$ equals $c^0_m$ on $B_m$ and the constant $c_\varnothing$ on $\Omega\setminus B_m$, so it is Borel, and $\lambda(U(c_m(z)))<1$ at every $z\in\Omega$ (by the case condition on $B_m$, by $\lambda(\varnothing)=0$ off it). This is (P3) on all of $\Omega$; by W1 the truncation is inactive on $Z$, so (P3) and (P4) coexist.

W3. Closing the quantifiers. Fix a name $\dot{\mathcal A}$ and let $\varphi$ be the implication in (5.1). If the top condition did not force $\varphi(\dot{\mathcal A})$, the Boolean value of its negation, a condition $q\ne0$, would force that $\dot{\mathcal A}$ is a family with $\forall y\,\lambda^*(\dot A_y)<1$ and $\neg\mathrm{Prof}(\dot{\mathcal A})$; $q$ then satisfies (5.2), the argument with $q$ in place of $p$ yields $q\Vdash\mathrm{Prof}(\dot{\mathcal A})$, and a nonzero condition forces a statement and its negation, which is impossible. So $\|\varphi(\dot{\mathcal A})\|=1$ for every name, and the value of $\forall\mathcal A\,\varphi(\mathcal A)$, the infimum over names, is $1$. The step "every generic $G\ni p$ satisfies $\mathrm{Prof}(\dot{\mathcal A}^G)$, hence $p\Vdash\mathrm{Prof}(\dot{\mathcal A})$" is the forcing theorem (see F3), used by the source in the same silent way.

Strongest attack

The attack aimed at W1, the only step where three separately fixed objects must agree: the enumeration of $D$ that identifies $2^D$ with $2^\omega$ in (5.10), the enumeration of $D_\alpha$ that identifies $2^{D_\alpha}$ with $2^\omega$ in (5.3), and the bijection $\pi_\alpha$. If the enumeration used in (5.10) were any enumeration of $D$ other than the $\langle d_n\rangle$ that $\pi_\alpha$ carries to $\langle(\alpha,n)\rangle$, then $z_\alpha\restriction D$ would be $\dot r_\alpha^G$ composed with a nontrivial permutation of $\omega$; the binary-expansion map $\rho$ does not commute with permutations of coordinates, so $x_m(z_\alpha)\ne\dot x_{\alpha,m}^G$ in general, (5.4) would say nothing about $A_{x_m(z_\alpha)}$, and (P4) would fail. The source leaves this coherence implicit ("carrying the fixed enumeration of $D$ to $\langle(\alpha,n):n<\omega\rangle$", p. 5, and (5.10) on p. 7). The page closes it explicitly: its Definitions identify $2^D$ with $2^\omega$ "through the enumeration $\langle d_n\rangle$ of $D$ from Proposition 4.4", the Proposition 4.4 page's interface gives $\pi_\alpha(d_n)=(\alpha,n)$, and the (P4) paragraph computes $(z_\alpha\restriction D)(d_n)=G(\alpha,n)=\dot r_\alpha^G(n)$. The attack fails.

A second attack on the same step: the reading (4.2) is a statement about the mixed names, while (5.4) is a statement about the names the maximum principle produced. Below $p$ the two agree, and (5.4) is used only in $M[G]$ with $p\in G$, so the values coincide; the attack fails. A third, on (P2): had $\rho$ been the plain binary-expansion map into $[0,1]$, the all-ones sequence would send $x_m$ to $m+1\notin I_m$ and the codomain in (P2) would fail on a null set; the page's instance redefines $\rho$ as $0$ on the eventually-one sequences, which keeps the range in $[0,1)$, leaves the pushforward Lebesgue measure on $[0,1)$ (a null set was changed), keeps $\rho$ Borel, and leaves every fiber countable (the fiber of $0$ is the eventually-one sequences with the zero sequence; every other fiber is a singleton, a dyadic rational keeping only its expansion ending in zeros). Then, for Borel $B\subseteq\mathbb R$, $\nu(x_m^{-1}(B))=\mu_D(\rho^{-1}(B-m))=\lambda((B-m)\cap[0,1))=\lambda(B\cap I_m)$, using that restriction to $D$ carries $\nu$ to the fair-coin measure $\mu_D$ and translation invariance. The attack fails.

Premises

• Definition 3.1 (source p. 3, displays (3.1)--(3.4); the Theorem 3.2 page). Interface: (P1)--(P4) as restated above, with $\nu^*$ the infimum over Borel supersets and the coding space carrying Borel $x\in U(c)$ and $c\mapsto\lambda(U(c))$ and a code $c_\varnothing$ of the empty set. Held; read clause by clause against page image 3. The page uses exactly these four clauses.
• Proposition 4.4 (source pp. 5--6, display (4.2); the Proposition 4.4 page). Interface: provable in ZFC + CH; for $p\in\mathbb B(\Theta)$, a standard Borel $X$ and names $\dot w_\alpha$ ($\alpha<\kappa$) for elements of $X$, there are $J\subseteq\kappa$ of size $\kappa$, a countable root $R$ supporting $p$, pairwise disjoint countable petals $P_\alpha\supseteq D_\alpha$ ($\alpha\in J$), a countable pair $D\subseteq P$ with enumeration $\langle d_n\rangle$ and bijections $\pi_\alpha\colon P\to P_\alpha$ with $\pi_\alpha(d_n)=(\alpha,n)$, and one Borel $F\colon2^R\times2^P\to X$ with $\Vdash\dot w_\alpha=F(\dot G\restriction R,(\dot G\restriction P_\alpha)\circ\pi_\alpha)$ for $\alpha\in J$. Held; read against page image 5 (statement) and the page as of that time. Applied with $X=\mathcal O^{\mathbb Z}$, a countable product of standard Borel spaces, hence standard Borel, and with names that the top condition forces into $X$ after the mixing; its hypotheses are met. Standing: imported as an author-recorded reconstruction of a ZFC + CH result; CH enters the page only here.
• Lemma 4.5 (source p. 6, display (4.3); the Lemma 4.5 page). Interface: provable in ZFC; for an uncountable family of pairwise disjoint countable petals each identified with a fixed countable $P$ and any further coordinate set $\Gamma$ disjoint from them, $\mathbb B(\bigcup P_\alpha\cup\Gamma)$ forces $\nu^*(\{\dot z_\alpha:\alpha\in J\})=1$ for the normalized generic points $\dot z_\alpha=(\dot G\restriction P_\alpha)\circ\pi_\alpha$. Held; read against page image 6. Applied with the petals and bijections of Proposition 4.4 and $\Gamma=\Theta\setminus\bigcup P_\alpha$, so the algebra is $\mathbb B(\Theta)$; $J$ has size $\omega_2$ in the ground model, which is the uncountability the lemma needs.
• Conventions (R1)--(R5) of the Lemma 4.1 page, assumed by the Standing paragraph: countable supports, the canonical generic point $u_G$ and the value of $\dot G\restriction S\in W$, the maximum principle with mixing along a partition of unity, absoluteness of Borel codes and Borel statements, and Borel isomorphism. The page also uses the forcing theorem in the direction "if every generic $G\ni p$ gives $M[G]\models\varphi$ then $p\Vdash\varphi$", named on the page but stated nowhere among (R1)--(R5) (F3).
• Outer regularity of $\lambda^*$, used as a definition by declaration. It is equivalent to the interval-cover definition: given a cover by intervals of total length below one, enlarge the $n$-th to an open interval longer by $\varepsilon2^{-n-1}$; the union is open, contains the set, and has measure below one for small $\varepsilon$; conversely an open set is a countable disjoint union of open intervals whose lengths sum to its measure.
• The instance of $\rho$, labeled a compilation fill; verified above (Borel, range in $[0,1)$, pushforward Lebesgue, countable fibers).
• Standard facts used without citation: a section of a Borel map is Borel; a map agreeing with Borel maps on the pieces of a Borel partition is Borel; the marginal of a product measure on a subset of coordinates is the product measure there; a countable subset of $2^\omega$ is null; every open subset of $\mathbb R$ is the union of the rational intervals it contains (F4).

Findings

F1. Severity: note. Location: "The truncation is what makes (P3) hold on all of $\Omega$ rather than only on $Z$." Defect: the sentence paraphrases the source's reason with a different one. The source says (p. 7) "This Borel truncation is the reason no conditional-conullity argument is needed." Witness: without truncation, (4.2) and (5.4) give $p\Vdash\lambda(U(F_m(\dot G\restriction R,\dot z_\alpha)))<1$, so by Fubini the $g$-section of $\{(u,v):\lambda(U(F_m(u,v)))\ge1\}$ is $\nu$-null for almost every $g$ below $p$; a conditional-conullity argument would extend (P3) from $Z$ to a $\nu$-conull set, still not to all of $\Omega$, and a modification on the null remainder would still be needed. The page's sentence is true of what (5.4) directly gives but misdescribes the alternative the source names. Proposed replacement: "The truncation makes (P3) hold at every $z\in\Omega$ outright; without it one would have to show that the section at the generic $g$ of the Borel set of pairs $(u,v)$ with $\lambda(U(F_m(u,v)))\ge1$ is $\nu$-null (the source's 'conditional-conullity argument') and then still modify $c^0_m$ on that null set."

F2. Severity: note. Location: "In $M[G]$, since $p\in G$ and $\dot w_\alpha$ is read by $F$, $\dot c_{\alpha,m}^G=F_m(g,z_\alpha)$". Defect: the equality follows from (4.2), which the top condition forces; $p\in G$ is not needed for it and is used only two sentences later for (5.4). Witness: source (4.2), p. 5, is stated under $\Vdash_{\mathbb B(\Theta)}$ with no condition. Proposed replacement: "In $M[G]$, since $\dot w_\alpha$ is read by $F$ under the top condition, $\dot c_{\alpha,m}^G=F_m(g,z_\alpha)=c^0_m(z_\alpha)$; and since $p\in G$, this value satisfies (5.4)."

F3. Severity: suggested. Location: "so $p\Vdash\mathrm{Prof}(\dot{\mathcal A})$ by the forcing theorem" in Closing the quantifiers, and the passage "In any extension by a generic containing $p$ ... By the maximum principle (R3)" in Names for envelopes. Defect: the direction of the forcing theorem used, from truth in every generic extension containing $p$ to $p$ forcing, is named but appears nowhere among the assumed conventions: (R3) on the Lemma 4.1 page carries the heading "Forcing theorem and maximum principle" but states only the maximum principle and mixing. Witness: the Lemma 4.1 page as of that time, item (R3); the source uses the same step silently ("Since $p$ and $G\ni p$ were arbitrary, (5.1) follows", p. 7). Proposed replacement: add to Standing "The forcing theorem is used in the direction: if every generic $G\ni p$ satisfies $\varphi$ in $M[G]$, then $p\Vdash\varphi$; equivalently, a condition not forcing $\varphi$ has an extension forcing $\neg\varphi$ (T. Jech, Set Theory, third millennium edition, Chapter 14)", or point to the Theorem 1.1 page's import of the forcing theorem.

F4. Severity: note. Location: "and every open set has a code" in Names for envelopes. Defect: the coding space of the Theorem 3.2 page is declared onto the open sets in the ground model; the page uses surjectivity inside $M[G]$ for the reinterpreted coding. Witness: for the standard coding fixed on the Theorem 3.2 page (codes as sets of rational intervals) this holds in every model, since an open set is the union of the rational intervals it contains, but it is not a consequence of the two Borel properties alone. Proposed replacement: "and every open set has a code (for the standard coding, an open set is the union of the rational intervals it contains, in $M[G]$ as in $M$)".

F5. Severity: suggested. Location: the Standing paragraph, "The instance of the map $\rho$ under Definitions is a compilation fill." Defect: three further passages expand one-sentence remarks of the source and are not marked as expansions: the Closing the quantifiers paragraph (source: "Since $p$ and $G\ni p$ were arbitrary, (5.1) follows", p. 7), the (P2) computation (source: "Then $x_m$ has Lebesgue distribution on $I_m$", p. 7), and the verification $x_m(z_\alpha)=\dot x_{\alpha,m}^G$ (source: "By (4.2) and (5.4)", p. 7). Each expansion is correct (W1, W3 and the Strongest attack section); the finding is one of labeling only. Proposed replacement: append to Standing "The verification of (P2), the identity $x_m(z_\alpha)=\dot x_{\alpha,m}^G$ and the closing paragraph expand one-sentence remarks of the source."

Verdict

Source fidelity: faithful. The Statement is the source's (5.1) verbatim in content; the hypotheses, conclusion, quantifiers and the convention on $\rho$, the identifications and the pullback are those of the source or are declared where they differ in notation; the locators (physical pp. 6--7, labels (5.1)--(5.11)) are right; the only supplied object, the instance of $\rho$, is labeled.

The argument as reconstructed: sound. Every deduction was re-derived (W1 to W3 and the Checklist); the imported results are applied inside their hypotheses at the strength their own pages state.

Limitations: Proposition 4.4, Lemma 4.5 and Definition 3.1 are consumed through their interfaces and are not re-proved here; the conventions (R1)--(R5) and the forcing theorem are standard imports taken on faith; the source's own text of Theorem 5.1 was checked only as far as the reconstruction tracks it; and the review touches no Lean development and no computation. This focused review assigns no tier and changes no status.